ISSN 0001-3765 www.scielo.br/aabc
On four dimensional Dupin hypersurfaces in Euclidean space
CARLOS M. C. RIVEROS and KETI TENENBLAT Departamento de Matemática, IE.UNB, 70910-900 Brasília, Brazil
Manuscript received on August 8, 2002; accepted for publication on October 28, 2002; contributed byKeti Tenenblat*
ABSTRACT
Dupin hypersurfaces in five dimensional Euclidean space parametrized by lines of curvature, with four distinct principal curvatures, are considered. A generic family of such hypersurfaces is locally characterized in terms of the principal curvatures and four vector valued functions of one variable. These functions are invariant by inversions and homotheties.
Key words:Dupin hypersurfaces, Laplace invariants, lines of curvature, principal curvatures.
1 INTRODUCTION
Dupin surfaces were first studied by Dupin in 1822 and more recently by many authors (Riveros and Tenenblat preprint, Cecil and Chern 1989, Cecil and Jensen 1998, 2000, Miyaoka 1984, Nieber-gall 1992, Pinkall 1985a,b, Stolz 1999 and Thorbergsson 1983), which studied several aspects of Dupin hypersurfaces. The class of Dupin hypersurfaces is invariant under conformal transfor-mations. Moreover, the Dupin property is invariant under Lie transformations (Pinkall 1985b). Therefore, the classification of Dupin hypersurfaces is considered up to these transformations. The local classification of Dupin hypersurfaces is considered up to these transformations. The local classification of Dupin surfaces inR3is well known. Pinkall (1985a) gave a complete classification up to Lie equivalence for Dupin hypersurfacesM3 ⊂ R4. However, the classification of Dupin hypersurfaces for higher dimensions is far from complete. Therefore, it is important to characterize such submanifolds. Our main result, Theorem 5, provides a local characterization of generic Dupin hypersurfaces inR5, with four distinct principal curvatures. The details of the proof can be found in (Riveros and Tenenblat preprint) and they will appear elsewhere. The proof is based on the theory of higher-dimensional Laplace invariants, which we recall in section 2, and the properties of Dupin hypersurfaces with distinct principal curvatures, given in section 3.
*Member of Academia Brasileira de Ciências Correspondence to: Keti Tenenblat
2 THE HIGHER-DIMENSIONAL LAPLACE INVARIANTS The results in this section were obtained by Kamram and Tenenblat (1996, 1998).
We consider linear systems of second-order partial differential equations, of the form
Y,kl+aklkY,k+akll Y,l+cklY =0, 1≤k=l≤n, (1)
whereY is a scalar function of the independent variablesx1, x2, . . . , xn, Y,ldenotes the derivative ofY with respect toxl and the coefficientsa andcare smooth functions ofx1, x2, . . . , xn which are symmetric in the pair of lower indices and satisfy certain compatibility conditions. The general form of the system (1) is preserved underadmissible transformations
Y = ϕ(x1, x2, . . . , xn)Y ,¯
x1 = fi(x¯i), 1≤i≤n,
(2)
whereϕis smooth and non-vanishing and thefi’s are smooth and have non-vanishing derivatives. It is easily verified that under an admissible transformation, the coefficientsa and c transform according to,
¯
allk = fk′akll +(logϕ),k
,
¯
clk = fl′f
′
k
clk+allk(logϕ),l+alkk(logϕ),k+ ϕ,kl
ϕ
, 1≤k=l≤n,
and the system (1) is
¯
Y,kl+ ¯aklkY¯,k+ ¯akll Y¯,l+ ¯cklY¯ =0, 1≤k=1≤n.
Thehigher-dimensional Laplace invariantsof (1) are defined to be then(n−1)2functions given by
mij =aiij,i+a i ija
j
ij −cij, mij k =akjk −a i
ij, k=i, j , (3)
for all ordered pairs(i, j ),1≤i=j ≤n.
Lemma 1. The higher-dimensional Laplace invariants of a compatible system (1) satisfy the following relations:
mij k+mkj i = 0, mij k,k−mij kmj ki −mkj = 0, mij,k+mij kmik +mikjmij = 0, mij k−mij l−mlj k = 0, mlik,j +mij lmkil+mlj kmkij = 0.
(4)
The functionsmij, mij k are invariant under pure rescalings (2). The expression of a system (1) in terms of its higher-dimensional Laplace invariants is established in the following Theorem. For proof and details we refer also to (Tenenblat 1998).
Theorem 2. Given any collection of n(n−1)2, n ≥ 3, smooth functions of x1, x2, . . . , xn mij, mij k,1 ≤ i, j, k ≤ n, i, j, k distinct, satisfying the constraints (4), there exists a linear system(1) whose higher-dimensional Laplace invariants are the given functionsmij, mij k. Any such system is defined up to rescaling(2). A representative is given by
Y,ij+AY,j−mijY = 0, Y,ik+(A+mj ik)Y,k−mikY = 0, Y,j k+mikjY,j+mij kY,k = 0, Y,lk+miklY,l+milkY,k = 0.
where (i, j ) is a fixed (ordered) pair, 1 ≤ i, j, k, l ≤ nare distinct and A is a function which satisfies the following:
A,j =mj i−mij, A,k = −mj ki,i
3 DUPIN HYPERSURFACES WITH DISTINCT PRINCIPAL CURVATURES
Definition3. An immersionX:⊂Rn →Rn+1is called aDupin hypersurfaceif along each curvature line the corresponding principal curvature is constant.
LetX: ⊂ Rn → Rn+1be a Dupin hypersurface parametrized by lines of curvature, with distinct principal curvaturesλi,1≤i ≤nand leN:⊂Rn →Rn+1be a unit vector field normal toX. Then
X,i, X,j = δijgii, 1≤i ≤n,
N,i = λiX,i, 1≤i ≤n,
λi,i = 0, 1≤i≤n.
Moreover
X,ij −ŴiijX,i−Ŵ j
ijX,j =0, 1≤i =j ≤n,
Ŵiji = λi,j λj−λi
, 1≤i =j ≤n, (5)
whereŴk
ij are the Christoffel symbols.
As a consequence of the relations (3) and Lemma 1, we obtain the following expressions for the higher-dimensional Laplace invariants
mij = −Ŵij,ii +Ŵ i ijŴ
j ij,
Moreover, for 1≤i, j, k, l ≤n , i, j, k, ldistinct, one has
mij = 0, mij k+mkj i = 0, mij k,k−mij kmj ki = 0, mij k−mij l−mlj k = 0, mlik,j +mij lmkil+mlj kmkij = 0.
(6)
LetX:⊂Rn →Rn+1, n≥3, be a Dupin hypersurface parametrized by lines of curvature, with distinct principal curvatures. Consider a homothety
D:Rn+1 → Rn+1
X → D(X)=aX, a∈R, a=0
and assuming 0∈/X()an inversion
In+1:Rn+1− {0} → Rn+1− {0}
X → In+1(X)= X X, X.
LetY =In+1(X)andY¯ =D(X). Then the higher-dimensional Laplace invariants ofY andY¯ are those ofX, since inversions and homotheties are rescalings (2).
The next result is an application of Theorem 2.
Lemma 4. LetX: ⊂ Rn → Rn+1, n ≥ 3, be a Dupin hypersurface parametrized by lines of curvature and λr,1 ≤ r ≤ n, distinct principal curvatures of each point. For i, j, k fixed, 1≤i=j =k≤n, the transformation
X=VX,¯ with V = e
λk−λj λj−λimj kidxk
λj −λi
, (7)
transforms the system(5)into
¯
X,ij+AX¯,j = 0,
¯
X,ir +(A+mj ir)X¯,r = 0,
¯
X,j r +mirjX¯,j+mij rX¯,r = 0,
¯
X,rl+milrX¯,r +mirlX¯,l = 0,
wherel=r are distinct fromi, j,
A= −
mj ki,idxk.
and
4 A CHARACTERIZATION OF DUPIN HYPERSURFACES INR5
We can now state our main result which provides a local characterization for generic Dupin hy-persurfaces inR5, with four distinct curvatures. Considering the functionsm
ij k satisfying (6), we introduce the following functions admitting thatm213=0, m214=0 andm314=0,
T = m214+
log m213 m314 ,1 (8)
U = m314+
log m213 m314 ,1 (9)
P14=m213T , P24=m213U, P34=m214U. (10)
Theorem5. LetX:⊂R4→R5, be a Dupin hypersurface parametrized by lines of curvature, with principal curvatures,λi,1 ≤ i ≤ 4, distinct at each point. Supposem234 = 0,m423 = 0, P4
i =0, for alli,1≤i≤3, then
X = V 3
i=1
(−1)i+1Bi4
, (11)
where
V = e
λ3−λ2
λ2−λ1m231dx3
λ2−λ1
, Bi4 = 1 Q4
i
Q4
iG1(x1) P4
i
dx1+Gi+1(xi+1)
, 1≤i≤3, (12)
Pi4,1 ≤ i ≤ 3, are defined by(10),Gi(xi),1 ≤ i ≤ 4, are vector valued functions ofR5, A = − m231,1dx3and
Q4i =
eAdx1 if i =1
e(A+m21(i+1))dx1 if 2≤i ≤3.
(13)
Moreover, considering
α1=
A+ λ2,1 λ1−λ2
M+M,1, αj = λ1,j λj −λ1
M+M,j, 2≤j ≤4, (14)
where M =
3
i=1
(−1)i+1Bi4, the functions Gi(xi) satisfy the following properties in , for all
1≤i, j ≤4:
a) αi =0,
c) λi = −
α,ii, α1×α2×α3×α4
V |αi|2|α1| |α2| |α3| |α4|.
Conversely, letλi: ⊂ R4 → R, 1 ≤ i ≤ 4 be real functions distinct at each point, such that λi,i =0, and the functionsmij k, defined by
mij k = λi,j λj −λi
− λk,j λj−λk
, 1≤i =j =k≤4.
satisfy(6). Then for any vector valued functions Gi(xi) satisfying the properties a)b)c) above, whereαi are defined by(14), the applicationX:⊂R4 →R5given by(11)describe a Dupin hypersurface parametrized by lines of curvature whose principal curvatures areλi.
Sketch of the proof
LetXbe a Dupin hypersurface as in Theorem 5 then it follows from Lemma 4 that
X = V
m213
W3−W2
,
whereV is given by (12) and the vector valued functionsW3(x1, x2, x4)andW2(x1, x3, x4)satisfy
systems of differential equations of Laplace type. The solutions of the systems are given by
W3 = m213 Q4
1
Q4
1G1(x1) P4
1
dx1+G2(x2)
−m314 Q4
3
Q4
3G1(x1) P4
3
dx1+G4(x4)
W2 = m213 Q42
Q4
2F (x1) P24
dx1+G3(x3)
−m214 Q43
Q4
3F (x1) P34
dx1+ ¯G4(x4)
It can be shown that
G1(x1)=F (x1), G4(x4)= ¯G4(x4)
which proves (11).
The conditions a), b) of Theorem 5 are obtained considering that the vectorsX,iare orthogonal and non-vanishing, condition c) is equivalent to requiringλi to be principal curvature ofX. The converse is a straightforward calculation.
Remark. One can show that the vector valued functionsGi(xi) in Theorem 5 are invariant by
inversions and homotheties of the corresponding Dupin hypersurfaces in R5. We conclude by
mentioning that a characterization of a non-generic family of Dupin hypersurfaceMn⊂Rn+1, n≥
3 whose principal curvatures are distinct andmij k =0, ∀i, j, kdistinct has been obtained and it will appear elsewhere.
RESUMO
Consideramos hipersuperfícies de Dupin no espaço Euclideano de dimensão cinco, parametrizadas por
li-nhas de curvaturas, com quatro curvaturas principais distintas. Uma família genérica de tais hipersuperfícies
é localmente caracterizada em termos das curvaturas principais e quatro funções vetoriais de uma variável.
Essas funções são invariantes por inversões e homotetias.
Palavras-chave: hipersuperfícies de Dupin, invariantes de Laplace, linhas de curvatura, curvaturas
princi-pais.
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